by David Orrell
Pythagoras is credited with a number of mathematical discoveries, including the properties of what are now called the Platonic solids—the pyramid-shaped tetrahedron, cube, octahedron, dodecahedron, and icosahedron. (Every face of these polyhedron figures is identical, and remarkably only five exist.) However, he is best known for his famous theorem, which states that in a right triangle, the square of the side opposite the right angle equals the sum of squares of the other two sides. While the Egyptians and Babylonians were probably aware of this relationship well before Pythagoras, at least for certain triplets, the Pythagoreans appear to have been the first to generalize the concept. Just as the theory of musical harmony applies to any instrument, the Pythagorean theorem applies to any right triangle. As the Pythagoreans understood, the power of mathematics comes from knowing that a single law holds in all cases—from reducing the plurality to one. The theorem is still one of the most important results in mathematics, and it’s used in everything from engineering to nuclear physics.
FIGURE 1.2.The Theorem Of Pythagoras: A2+B2+C2
The Pythagoreans believed, almost as an article of religious faith, that the world was made up of positive integers and their ratios, such as the fraction ¾, which are called rational numbers. Ironically, Pythagoras’s theorem about triangles led to the discovery of numbers that cannot be expressed as a ratio. A right triangle with two sides both equal to one unit has a hypotenuse (from the theorem) of the square root of two. Hippasus, one of the Pythagoreans, showed that the root could not be expressed as a ratio of two integers. In other words, it was irrational. His comrades could not accept that such a number existed: it was as if someone had found a bum note in the music of the spheres. Hippasus made the mistake of publicizing the results openly, “to the profane and to those . . . without disciplines and theories.”22 He died shortly afterwards under mysterious circumstances. It was said that “the Divine Powers were so indignant that he perished in the sea.”23 (See notes for a proof that Hippasus was right.24)
It is strange that numbers that cannot be expressed as a ratio of whole numbers are called irrational, as if they were in some way deviant; there are far more of them than there are rational numbers, just as there are many more pitches of sound than those found on a keyboard. In fact, if you could choose only one number by throwing an imaginary dart at the interval from zero to one, the chances of hitting a rational number are zero.25 You might aim to hit ½, but you’d actually get some irrational number with an endless sequence of digits, like 0.5083428 . . .
The Pythagorean commune grew in both size and power, to the point where it exerted considerable influence over Croton and the surrounding area. It is even believed that Pythagoras became the local “master of the mint,” bringing the first metal coinage to the region.26 Eventually, though, this rational society became the victim of seemingly irrational forces. People—especially those who had been excluded from membership—began to speak against the secretive and elitist group. The citizens started to harass the Pythagoreans in the streets. When a number of them assembled at the house of Milon, an Olympic athlete, a mob surrounded them and set the house on fire; only two escaped the conflagration. In another incident, forty members of the group were attacked and killed. Pythagoras himself managed to escape, and probably died in exile. Even Apollo’s arrow was no protection against the madness of crowds; but the demise of the Pythagoreans marked only the first stage in the development of numerical prediction. As the novelist and philosopher Arthur Koestler wrote of Pythagoras: “His influence on the ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him.”27 He didn’t just predict the future; he also helped define it.
THE ACADEMICS
Since the Pythagoreans didn’t believe in recording their methods, our accounts come mostly from future documenters. One of these was a man born, it is said, with the name Aristocles. His mother and father came from famous, wealthy families in Athens. His uncle was a friend of the philosopher Socrates. Perhaps because of his physical bulk—he was a trained wrestler—or the width of his forehead, he was usually known by his nickname, which roughly translates to “the broad.” He was Plato.
Like Pythagoras, Plato was a man of many talents. He knew the arts of politics, philosophy, and war. He was also a playwright. According to Diogenes, he began his career as a writer of tragedies. After hearing Socrates talk, however, he gave up on the theatre, and even set fire to a play that he had been planning to enter into a drama competition. Instead, he began pouring his creative energy into the writing of philosophical dialogues. While these weren’t plays, they did show his enormous skill at crafting entertaining dialogue. They also often blur the line between fiction and nonfiction. It is hard to know whether the dialogues represent things that were actually said or are a fictionalized account. Or whether they even represent what Plato himself thought on a subject.
An example of this ambiguity is to be found in Plato’s Defence of Socrates. Socrates, the son of a sculptor, was another servant of Apollo who dedicated his life to understanding the causes that underlie the universe. This quest took on a new form after his friend Chaerephon visited the Delphic Oracle and asked whether there was anyone who was wiser than Socrates. The Pythia replied that there was not. Since Socrates was unaware of any wisdom within himself—he often joked that “the only thing I know is that I know nothing”—he interpreted this to be a mission from Apollo to visit those who claim to be wise and discover their wisdom. He therefore went to poets, artisans, and statesmen across the land, but after closely questioning them, he realized that they were not wise at all. “And so I go my way, obedient to the god, and make inquisition into the wisdom of anyone, whether citizen or stranger, who appears to be wise; and if he is not wise, then in vindication of the oracle I show him that he is not wise; and this occupation quite absorbs me, and I have no time to give either to any public matter of interest or to any concern of my own, but I am in utter poverty by reason of my devotion to the god.”28
Needless to say, this attitude annoyed a lot of people. In 423 B.C., Aristophanes wrote a comedy called The Clouds, in which the main character, also called Socrates, worships clouds and other natural phenomena rather than the gods. The play was produced in a competition at the Great Dionysia. It came third out of three plays but was published a few years later. The Greeks were avid theatre-goers, and the play turned Socrates into first a figure of fun, then a tragic anti-hero.
Twenty-four years after The Clouds was produced, Socrates found himself on trial, accused of believing in new divinities and corrupting the youth. Perhaps his accusers were confusing the character in the play with the real person. Plato’s Defence of Socrates is a dramatic account of how the philosopher defended himself against the citizens of Athens. It is not known whether it is a verbatim transcript of what Socrates said, a heavily edited version, or a fictionalization that is really Plato’s defence of his friend and mentor. Again, the story and the reality are hard to separate.
In any case, according to Plato, the jury was not convinced by Socrates’ case. The prosecutors were seeking the death penalty, but Socrates was given the chance to offer an alternative. He first suggested free meals for himself for life. This didn’t go down well, so he suggested a nominal fine of one mina. As Socrates dug a deeper and deeper hole for himself, his friends, including Plato, offered to pay a more substantial fine. But it was to no avail, and Socrates was put to death by poisoning with hemlock.
The death of Socrates was not in vain, for it had a huge impact on Plato. Sick of the politics of Athens, he travelled to Egypt, Sicily, and Italy. It was in Italy that he learned of the work of Pythagoras and met his disciples. From them, according to the scholar G.C. Field, he formed the idea “that the reality which scientific thought is seeking must be expressible in mathematical terms, mathematics being the most precise and definite kind of thinking of which we are capable.”29 The only way to overcome the ignorance that Socrates had expo
sed was with numbers.
When Plato returned to Athens, around 387 B.C., he established what became the longest-running learning institution in the history of mankind—the precursor to today’s universities. The Academy, so named because the land belonged to a man called Academos, was dedicated to research and instruction in philosophy and science. Over the door was written, “Let no one unversed in geometry enter here.” Plato’s concentration on precise definitions, clear statement of hypotheses, and rigorous proofs of mathematical conjectures all prepared the ground for the major mathematical developments of ancient Greece, which underpin modern science. The Academy survived more than 900 years, until the Christian emperor Justinian, claiming it was a pagan establishment, closed it down in 529 A.D.
Students at the Academy would spend ten years studying the sciences of astronomy and mathematics, then five years studying dialectic (the art of posing and answering questions). Plato believed that dialectic was the path to wisdom, and through his dialogues, he contributed to the theory of arts from poetry to epistemology. He taught that material objects were imperfect versions of underlying forms, which existed in a static way, independent of time and space. An example of a perfect form was a mathematical object such as a line. A material manifestation of a line, such as a line drawn in the sand, was only a flawed reproduction of the real thing, like a poor photocopy. Every object had an associated form, which it yearned to be but could never reach. The plurality of different tables, for example, all aspired to the one true table. To Plato, the ultimate reality was not the chaotic, imperfect world that we see and hear and taste, but rather the abstract, eternal world of pure forms. Our world was just a blurred shadow of the real thing.
MATHEMATICAL BIOLOGY
In 430 B.C., the citizens of Athens were struggling with the real-world problem of infectious disease. Thucydides gave a graphic description of the plague that was afflicting the city. People were first attacked by “violent heats in the head, and redness and inflammation in the eyes,”30 along with sneezing, hoarseness, and a cough.“Discharges of bile of every kind named by physicians ensued, accompanied by very great distress.” The skin was “reddish, livid, and breaking out into small pustules and ulcers. But internally it burned so that the patient could not bear to have on him clothing or linen even of the very lightest description; or indeed to be otherwise than stark naked. What they would have liked best would have been to throw themselves into cold water; as indeed was done by some of the neglected sick, who plunged into the rain-tanks in their agonies of unquenchable thirst; though it made no difference whether they drank little or much.” In most cases, the disease proved fatal after seven or eight days. Some, like Thucydides himself, survived but were often maimed or blinded.
Near the height of the plague, a delegation was sent to Delphi to ask the oracle how it could be stopped. The oracle’s reply was that the altar of Apollo on the island of Delos, which was in the form of a perfect cube, should be doubled in size. In response, the delegates arranged for each edge of the cube to be extended by a factor of two; however, this increased the volume not by two but by eight. The oracle announced that Apollo—whose arrows were believed to cause plague sores—had been angered by this sloppy arithmetic, and indeed the outbreak grew worse.31 Plato was consulted. He told them, “The god has given this oracle, not because he wanted an altar of double the size, but because he wished in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt of geometry.” Only the magic of number could defeat the plague.
Perhaps he was right, because soon the plague began to ease, though not before claiming about a third of the population of Athens. The problem of how to double the cube didn’t go away, though. The Athenian mathematicians, at least those still surviving, believed that all mathematical problems could and should be solved using only a compass and a ruler, tools that corresponded to the perfect forms of circles and straight lines. Anything that couldn’t be expressed in these terms was out of bounds. Just as the Pythagoreans eventually encountered a problem that could not be solved by rational numbers, however, the Athenians found that many of their problems could not be solved with these two tools alone. They could not construct a square with the same area of a given circle or double a cube or trisect an angle into three equal angles. Their insistence on static forms stood solidly in the way of progress.
Eventually, solutions for all these challenges were arrived at using so-called mechanical curves. These needed to be traced out by sliding lines around a point. Because they introduced the idea of change and motion—a bad thing in Pythagoras’s list—they were not considered to be real geometry.
THE GREEK CIRCLE MODEL
To Plato and other philosophers of his time, the universe was not a place of chaotic flux and change but a kind of endlessly repeating cycle. Like Pythagoras, Plato thought that time moves in circles. The future had already been determined, and it was the past. Because events did not occur randomly, but were known in advance, it followed that the future could be predicted. The logical place to start was up above, in the heavens.
The exemplars of circular, repetitive motion were the stars and the planets, which were believed to move in perfect circles around the earth. Indeed, careful observations of the stars showed that they did move in a circular fashion. The planets—in particular Mars, Jupiter, and Saturn—were more tricky. Their path around the night sky involved a fair amount of wandering (the word “planet” is from the Greek for wanderer) and even backtracking. It seemed they had a life of their own. For example, Mars advanced around the sky from west to east, completing a revolution in about 780 days; but partway around it would stop, backtrack a little, and then resume its forward motion. And 780 days later, it would do the same thing.
The wanderings of the planets seemed incompatible with simple circular motion, but Plato’s associate Eudoxus managed to come up with a model for the universe that captured such effects.32 Imagine the earth surrounded by a huge crystalline sphere that contains the stars. The sphere rotates around us once per day. Inside this sphere is a separate transparent sphere that contains the sun. It too rotates around the earth once a day, but it also rotates annually at an approximate 23.5-degree angle to the line joining the centre of the earth to the North Pole. The angle accounts for the seasons, since half the year one side of the globe will receive more sun, while the other half of the year it receives less (as shown in figure1.3 on page 42). This much was simple; the movements of the planets were more complicated. These were also modelled by spheres, whose axes of rotation were fixed to other spheres (which could themselves rotate). The resulting nest of twenty-seven rolling spheres was capable of producing highly complex motion. With carefully selected rates and angles of rotation, the model could adequately represent the motion of the heavens.
This model, which I will refer to as the first Greek Circle Model, was an amazingly ingenious geometrical accomplishment, and it can be viewed as a direct precursor to the mathematical models that are used today to simulate physical systems. Of course, it was purely descriptive, and was based on a hypothesis of circular motion, as opposed to rigorously derived physical laws of motion. The fact that it worked quite well as a model of the universe is a poignant reminder that a model that can be made to fit the data isn’t necessarily an accurate representation of reality.
FIGURE 1.3 The angle of the earth relative to the sun means that each hemisphere gets more sun in half of the year (its summer) and less in the other (winter). The angle actually varies from 21.8 to 24.4 degrees, returning to centre about every 42,000 years, owing to slight wobbles in the planetary system.
THE WORLD’S TUTOR
The Academy was the elite institution, the Ivy League or Oxbridge of its time, and many of Plato’s students went on to make major contributions to Greek science and philosophy. His star student was Aristotle, who stayed at the Academy for twenty years, from the time he was eighteen until Plato’s death in 384 B.C. Aristotle then took a job w
ith King Philip of Macedonia, tutoring his son Alexander for three years. We are familiar with the work of Aristotle and Plato largely because of Alexander the Great. When Alexander went to Delphi to obtain his oracle, the Pythia refused. He kept insisting, and even threatened her with force. Finally, she told him, “You can do what you like.”33 This was like telling George W. Bush not to hold back so much, and Alexander went on to conquer the Middle East, Persia, and Egypt, as well as parts of Afghanistan, Central Asia, and India. The library in one of the cities named after him—Alexandria, in Egypt—became the major repository of Greek knowledge, and eventually cemented Aristotle’s position as tutor to much of the world.
In 355 B.C., Aristotle returned to Athens and set up his own institute, called the Lyceum after the old temple of Apollo in which it was located. Like the Academy, the Lyceum taught a range of subjects, such as politics, ethics, and science. While Plato was fascinated by the abstract properties of forms, Aristotle’s science was more grounded in observation of physical and natural phenomena. For example, he collated descriptions of about 500 different types of animals, many of which he dissected. In Raphael’s painting The School of Athens, Plato is shown gesturing to the heavens while Aristotle is lowering his hand to the ground, as if to bring the theorist back to earth.